Why can't I see this curve's vertices?
Don't know of any shortcut to show and hide curve handles, or a way to turn them off by default in the settings (are you sure you didn't just add a mesh circle until now?).

But you can turn handles on or off through the Properties Panel (N) > Curve Display > Handle.



By the way, just for a bit of terminology and trivia:
These points on bezier curves are called Control Points, rather than Vertices (which are mesh points).